Indeed, the value of the integral for any \(D_{\alpha }\)  is uniquely determined by the adelic norm \([\alpha O_K: \mathbb {Z}]=\frac{\mu _{K}(\alpha O_L)}{\mu _{K}(O_L)}=|\alpha |\)  (see [1]}) and the measure \(\mu _{L/K}(O_L)\) . Note that the adelic norm is a product of local norms (see [2]} Lemma 4.1.2.) with the same relation with the local Haar measure \(\frac{\mu _{K_v}(\alpha O_{K_v})}{\mu _{K_v}(O_{K_v})}=|\alpha |_v\) . Hence, one sees that the value of \(\int _{\mathbb {A_K}} f_{\alpha }\)  is equal to \(\int _{\mathbb {A_K}} f_{1}\)  scaled by the adelic absolute value \(|\alpha |\)  of \(\alpha \) .
